Theorem nnssn0s 28251

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28246	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4085	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3977	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3895   ⊆ wss 3898  {csn 4575   0_s c0s 27767  ℕ_0scnn0s 28243  ℕ_scnns 28244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-ss 3915  df-nns 28246

This theorem is referenced by:  nnssno  28252  nnn0s  28257  nnn0sd  28258  nnsgt0  28268